Anisotropic Scaling Matrices and Subdivision Schemes
Mira Bozzini1, Milvia Rossini2, Tomas Sauer3 and Elena Volontè4
1 Università degli studi Milano-Bicocca, Italy, [email protected] 2 Università degli studi Milano-Bicocca, Italy, [email protected] 3Universität Passau, Germany , [email protected] 4 Università degli studi Milano-Bicocca, Italy, [email protected]
Abstract Unlike wavelet, shearlets have the capability to detect directional discontinuities together with their directions. To achieve this, the considered scaling matrices have to be not only expanding, but also anisotropic. Shearlets allow for the definition of a directional multiple multiresolution analysis, where perfect reconstruction of the filterbank can be easily ensured by choosing an appropriate interpolatory multi- ple subdivision scheme. A drawback of shearlets is their relative large determinant that leads to a substantial complexity. The aim of the paper is to find scaling matrices in Zd×d which share the properties of shearlet matrices, i.e. anisotropic expanding matrices with the so- called slope resolution property, but with a smaller determinant. The proposed matrices provide a directional multiple multiresolution anal- ysis whose behaviour is illustrated by some numerical tests on images.
1 Introduction
Shearlets were introduced by [6] as a continuous transformation and quite immediately became a popular tool in signal processing (see e.g. [7]) due to the possibility of detecting directional features in the analysis process. In the shearlet context, the scaling is performed by matrices of the form 4Ip 0 d×d M = D2S, D2 = ∈ R , 0 < p < d, (1) 0 2Id−p
1 and I ∗ S = p . (2) 0 Id−p
The diagonal matrix D2 results in the the anisotropy of the resulting shear- lets while the so-called shear matrix S, is responsible for the directionality of the shearlets. With a proper mix of these two properties, shearlets give a wavelet-like construction that is suited to deal with anisotropic problems such as detection of the singularities along curves or wavefront resolution [7, 8, 3]. In contrast to other approaches, such as curvelets or ridgelets, shearlets admit, besides a transform, also a suitably extended version of the multires- olution analysis and therefore an efficient implementation in terms of filter- banks. This is the concept of Multiple Multiresolution Analysis (MMRA) [8]. As for the classical multiresolution analysis, also the MMRA can be con- nected to a convergent subdivision scheme, or more precisely to a multiple subdivision scheme [13]. The refinement in a multiple subdivision scheme is controlled by choosing different scaling matrices from a finite dictionary (Mj : j ∈ Zs), where we use the convenient abbreviation Zs := {0, . . . , s−1}. If those scaling matrices are products of a common anisotropic dilation ma- trix with different shears, the resulting shearlet matrices yield a directionally adapted analysis of a signal. In particular, if we set Mj = D2Sj, j ∈ Zs, an n–fold product of such matrices, n M := Mn ··· M1 , ∈ Zs , allows to explore singularities along a direction directly connected to the sequence . In the shearlet case, this connection is essentially a binary expansion of the slope [8] while for more general matrices the relationship can be more complicated [2]. The key ingredient for the construction of a multiple subdivision scheme and its associated filterbanks [12] is the dictionary of scaling matrices. It turns out that, in order to have a well–defined (interpolatory) multiple sub- division scheme and perfect reconstruction filterbanks, we need to consider matrices (Mj : j ∈ Zs) that are expanding individually and jointly expanding considering them all together. Shearlet scaling matrices quite automatically satisfy these properties and they are also able to catch all the directions in the space due to the so-called slope resolution property. This means that any possible directions can be generated by M applied to a given refer- ence direction. Therefore, any transform that analyses singularities across this reference direction in the unsheared case, analyses singularities across arbitrary lines for an appropriate .
2 The drawback of the discrete shearlets is the huge determinant of the scaling matrix which is 2p+d, as minimum, and increases dramatically with the dimension d. The determinant of the scaling matrix gives the number of analysis and synthesis filters needed in the critically sampled filterbank, soit is related to the efficiency and the computational cost of any implementation. Therefore, it is worthwhile to try to reduce, or even better minimize it. The aim of this paper is to present a family of matrices with lower determi- nant that in any dimension d enjoys all the valuable properties of shearlets by essentially just giving up the requirement that the scaling has to be parabolic. This was also the aim of [2], where the authors present a set of matrices in 2×2 Z with minimum determinant among anisotropic integer valued matrices. d×d In contrast to that, here we consider matrices M ∈ Z which are the product of an anisotropic diagonal matrix of the form aI 0 D = p , a 6= b > 1, 0 < p < d, (3) 0 bId−p with a shear matrix. The choice to work with matrices of this form is motivated by the desire to preserve another feature of the shearlet matrices: the so-called pseudo-commuting property [1]. Even if this property is not necessary to construct a directional multiresolution analysis, it is very useful n to give an explicit expression for the iterated matrices M, ∈ Zs and the −1 d associated refined grid M Z . In this setting, the matrices with minimal determinant are 3I 0 D = p . (4) 0 2Id−p In this paper, we consider exactly this type of matrices and we prove that this new family has all the desired properties: they are expanding, jointly expanding, and provide the slope resolution property as well as the pseudo commuting property. Furthermore, it is possible to generate convergent multiple subdivision schemes and filterbanks related to these matrices. The paper is organized as follows. In Section 2, we recall some background material. Sections 2.2 and 2.3 introduce the concepts of MMRA and the related construction of multiple subdivision schemes and filterbanks for a general set of jointly expanding matrices. In Sections 2.4 we study the properties of shearlet scaling matrices and in Section 2.5 we recall some theorems by [1] that show which unimodular matrices pseudo commute with an anisotropic diagonal matrix in dimension d. This allows us to find pseudo- commuting matrices. Then, in Section 3, we introduce our choice of scaling
3 matrices for a general dimension d. These matrices are pseudo commuting matrices and we prove that they satisfy all the requested properties. Finally, in Section 4 few examples illustrate how our bidimensional matrices work on images.
2 Background material
In this section, we recall some of the fundamental concepts needed for MM- RAs.
2.1 Basic notation and definitions All concepts that follow are based on expanding matrices which are the key ingredient and fix the way to refine the given data, hence to manipulate d×d d them. A matrix M ∈ Z is called a dilation or expanding matrix for Z if all its eigenvalues are greater than one in absolute value, or, equivalently if kM −nk → 0 for some matrix norm as n increases. These properties suggest that | det M| is an integer ≥ 2. A matrix is said to be isotropic if all its eigenvalues have the same modulus and is called anisotropic if this is not the case. In this paper we consider a set of s expanding matrices, denoted by (Mj : j ∈ Zs) where Zs := {0, . . . , s − 1}. d d In order to recover the lattice Z from the shifts of the sublattice MZ , we have to consider the cosets
d d d d d Zξ := ξ + MZ := {ξ + Mk : k ∈ Z }, ξ ∈ M[0, 1) ∩ Z , where ξ is called the representer of the coset. Any two cosets are either d identical or disjoint, and the union of all cosets gives Z ; the number of cosets is | det M|. For the definition of a subdivision scheme, we have to fix some notation for infinite sequences. The space of real valued bi-infinite sequences indexedby d d d Z or, equivalently, the space of all functions Z → R is denoted by `(Z ). d By `p(Z ), 1 ≤ p < ∞, we denote the sequences such that 1/p X p kck d := |c(α)| < ∞, `p(Z ) α∈Zd
d d and use `∞(Z ) for all bounded sequences. Finally, `00(Z ) is set of sequences d from Z to R with finite support.
4 2.2 Multiple subdivision schemes and MMRA The concept of multiresolution analysis provides the theoretical background needed in the discrete wavelets context. Unlike curvelets and ridgelets, shearlets allow to define and use a fully discrete transform based oncas- caded filterbanks using the concept of MMRA. Here we recall the basic as- pects of MMRA and show when and how it is possible to generate a multiple multiresolution analysis with different families of scaling matrices. d d d A subdivision operator S : `(Z ) → `(Z ) based on a mask a ∈ `00(Z ) maps a sequence c ∈ `(Zd) to the sequence given by X Sc = a(· − Mα) c(α), α∈Zd −1 d which is then interpreted as a function on the finer grid M Z . An algebraic tool in studying subdivision operators is its symbol, defined as the Laurent polynomial whose coefficients are the values of the mask a,
∗ X α d a (z) = a(α)z , z ∈ (C \{0}) . α∈Zd As shown by [13], it is always possible to generate MMRA for an arbitrary family of jointly expanding scaling matrices (Mj : j ∈ Zs) in the shearlet context for an arbitrary dimension d, based on a generalization of subdivision schemes: multiple subdivision.
Definition 1 A multiple subdivision scheme S consists of a dictionary (Sj : j ∈ Zs) of s subdivision schemes with respect to dilatation matrices (Mj : d j ∈ Zs) and masks aj ∈ `00(Z ), j ∈ Zs, yielding the subdivision operators X Sjc = aj(· − Mjα) c(α), j ∈ Zs. α∈Zd n Given n ∈ N and ∈ Zs , = (1, . . . , n), the multiple subdivision operator takes the form
X d Sc = Sn ··· S1 c = a(· − Mα) c(α), a ∈ `00(Z ), α∈Zd where M := Mn ··· M1 .
In other words, the main idea of a multiple subdivision scheme is that in each step of the iterative process one can choose an arbitrary subdivision scheme from the family (Sj : j ∈ Zs).
5 Definition 2 The multiple subdivision scheme is called (uniformly) con- d vergent if for any infinite sequence : N → Zs of digits and any c : Z → ∞ d d R, c ∈ C (Z ) there exists a uniformly continuous function f,c : R → R such that −1 −1 lim sup Sn ··· S c − f,c M ··· M α = 0, n→∞ 1 1 n α∈Zd and f,c 6= 0 for at least one choice of c. Similar notions exist for convergence d in Lp(R ), 1 ≤ p < ∞ [13].
In multiple subdivision one considers all S simultaneously and a subdivision scheme is convergent if all possible combinations of the subdivision schemes converge (see for details [13, 12]. As a consequence, not only the individual matrix Mj, j ∈ Zs has to be expanding, but we need that they are jointly n expanding, i.e. M is an expanding matrix for any choice ∈ Zs , n ∈ N. While in the shearlets context this property follows directly from the particular choice of the Mj, j ∈ Zs, made there, for general families (Mj : j ∈ Zs) is an additional assumption that has to be verified in each concrete case. From now on, we assume that (Mj : j ∈ Zs) is a family of jointly expanding d×d matrices in Z . We recall the following generic construction for convergent multiple subdivision schemes that relies on the algebraic properties of the matrices Mj: given a scaling matrix Mj, j ∈ Zs, we follow [2] and consider the Smith factorization 0 Mj = ΘjΣjΘj, (5) 0 d 0 where Θj, Θj ∈ Z are unimodular matrices, i.e., | det Θj| = | det Θj| = 1, and Σj is a diagonal matrix with diagonal values σjk, k = 1, . . . , d. Ordering the diagonal values such as one values is divisible for the previous one, this can also be turned into a Smith normal form, see [9] for details. To construct a convergent subdivision scheme for Mj, we use a convergent scheme for Σj by means of a tensor product of univariate interpolatory schemes with arity equal to the respective diagonal values σjk and masks d bjk. The automatically convergent tensor product mask bΣj : Z → R is then of the form d d O Y d bΣj := bjk, bΣj (α) = bjk(αk), α ∈ Z , (6) k=1 k=1 so that the corresponding symbol can be written as d ∗ X α Y ∗ d bΣj (z) = bΣj (α) z = bjk(zi), z ∈ (C \{0}) . (7) α∈Zd k=1
6 Thus, the mask of the scheme related to Mj is
−1 aj := bΣj (Θj ·), and its symbol becomes
∗ X −1 α X Θj α ∗ Θj aj (z) = bΣj (Θj α)z = bΣj (α)z = bΣj (z ). (8) α∈Zd α∈Zd
In this way, the unimodular matrix Θj takes the role of a resampling op- d×d erator. Indeed, if Θ ∈ Z is any unimodular matrix, then, by the above reasoning, the resampled sequence c(Θ−1·) has symbol
∗ −1 ∗ Θ Θ θ1 θd c(Θ ·) (z) = c (z ), z := z , . . . , z , Θ = [θ1, . . . , θd], where θi, i = 1, . . . , d are the column vector of Θ. The following concept by [13] is a generalization of the well–known (z + 1)–factor of univariate subdivision that works as a multivariate smoothing operator.
Definition 3 If M = ΘΣΘ0 is a Smith factorization, the associated canon- ical factor with respect to M is the Laurent polynomial
d σ −1 d i zσiθi − 1 Y X θik Y ψM (z) := z = , (9) zθi − 1 i=1 k=0 i=1 where again θi are the column vectors of Θ and σi are the diagonal elements of Σ, i = 1, . . . , d.
Theorem 4 ([13]) The multiple subdivision scheme generated by (aj,Mj), ∗ d with Laurent polynomial aj = ψMj , j ∈ Zs, converges in L1(R ).
Since the autoconvolution of a compactly supported L1 function is continu- ous, the following conclusion can be done directly from Theorem 4.
Corollary 5 The multiple subdivision scheme generated by {(aj,Mj)} with a∗ = 1 ψ2 j ∈ j det Mj Mj , Zs, converges to a continuous limit function.
To each Mj, j ∈ Zs, we can associate a subdivision scheme by means of (8) by choosing a collection of univariate interpolatory subdivision schemes with arities σjk, k = 1, . . . , d, where the latter ones are the diagonal values of Σj in the Smith factorization of Mj. Considering the tensor product of
7 such schemes we get a the multiple subdivision scheme generated by Mj, j ∈ Zs. As an example, if bjk is the piecewise linear interpolatory scheme with arity σjk, it has mask 1 bjk = (0, 1, 2, . . . , σjk − 1, σjk, σjk − 1,..., 2, 1, 0) , σjk and its symbol is
z−(σjk−1) ∗ 2 σjk−1 2 bjk(z) = (1 + z + z + ... + z ) , k = 1, . . . , d. σjk
According to (7), the symbol of (6) takes the form 2 d −(σjk−1) σjk−1 ∗ Y zk X ` bΣj (z) = zi , (10) σjk k=1 `=0 hence, the symbol of the scheme associated to Mj (8) is
2 d σjk−1 z−(σjk−1)θk ∗ ∗ Θj Y X ` θk aj (z) = bΣj z = z . (11) σjk k=1 `=0
In this way we obtain a convergent multiple subdivision scheme.
n Proposition 6 The scheme S, ∈ Zs and n ∈ N, as constructed above, is a convergent multiple subdivision scheme that converges to a continuous function.
Proof. For each matrix Mj, j ∈ Zs, the corresponding canonical factor is d d d σjk−1 zσjkθk − 1 Y Y θk (σjk−1)θk Y X `θi ψMj (z) = = (1 + z + ... + z ) = z zσjk − 1 k=1 k=1 k=1 `=0 and since σ −1 2 1 d 1 jk 2 Y X `θk ψMj (z) = z det Mj σjk k=1 `=0
∗ −(σjk−1)θk is equal to aj up to the shift factors z , Corollary 5 yields that S, n ∈ Zs , n ∈ N, is convergent to a continuous limit function.
8 2.3 Filterbanks Filters, more precisely Linear and Time Invariant (LTI) filters [4], are linear operators on discrete signals that commute with translations and are basic building blocks of signal processing. Commuting with translations yields that they are stationary processes and can be represented by convolutions,
X d F c = f ∗ c = f(· − α) c(α), c : Z → R. α∈Zd
A filterbank consists of a finite set of analysis and synthesis filters, named fk and gk, k ∈ Zm, respectively, and operations to split a given signal into m subband components by means of filtering and downsampling, the so called analysis process, and to recombine these components into an output signal by upsampling and filtering, called the synthesis process (see [14]). The downsampling operator ↓M , respect to an expanding matrix M, is defined as d ↓M c = c(M·), c ∈ `(Z ).
To invert downsampling one introduces the upsampling operator ↑M
−1 d c(M α), α ∈ MZ , d ↑M c(α) = d c ∈ `(Z ). 0, α∈ / MZ ,
It is worthwhile to remark that only ↓M ↑M is an identity while ↑M ↓M is a lossy operator due to the decimation involved in the upsampling
d c(α), α ∈ MZ , ↑M ↓M c(α) = d 0, α∈ / MZ .
The expanding matrix M defines the decimation which is performed inthe downsampling after the convolution with the analysis filters. In the usual case of critically sampled filterbanks, the decimation rate | det M| coincides with the number m of filters in the filterbank in order to have thesame amount of information after decomposition. The filters f0 and g0 are nor- mally low-pass filters and all the other filters fk, gk with k 6= 0 are chosen as high-pass filters. Thus, the decomposition of a signal yields a coarse component, convolving with f0 and downsampling, and m − 1 details com- ponents given by the convolution of the signal with fk, k 6= 0, followed by downsampling. Hence, analysis computes the vector sequences
c 7→ (ck :=↓M fk ∗ c : k ∈ Zm)
9 while synthesis combines them into X gk ∗ (↑M ck) . k∈Zm In order to achieve perfect reconstruction, the usual standard assumption for a reasonable filterbank, the analysis and synthesis parts have to be the inverse of one another. This is equivalent to
X d c = gk ∗ (↑M ↓M fk ∗ c) , c : Z → R. k∈Zm
It was shown, for example by [12], that starting with an interpolatory con- vergent subdivision scheme it allows us to define filterbanks that achieve the perfect reconstruction by the prediction-correction method. A convergent interpolatory subdivision scheme with mask a takes coarse data and predicts how to refine them, thus it can be used as a synthesis filters g0 = a. In this case, the best way to define the low-pass analysis filter is toset f0 = δ, so the analysis process samples the initial data. Since, in general, prediction from decimation cannot reconstruct the signal, we introduce a correction f ∗(z) = zξk −a∗ (zM ) which defines the high-pass analysis filters as k −ξk , where d d ξk ∈ EM := M[0, 1) ∩ Z are the coset representers, k ∈ Zm \{0} and a−ξk the corresponding submask. This analysis can be complemented by a canonical synthesis process yielding the filters
∗ ∗ ξk M f0 (z) = 1, fk (z) = z − a−ξk (z ), ∗ ∗ ∗ −ξk (12) g0(z) = a (z), gk(z) = z , with ξk ∈ EM , k 6= 0. If we have not a single expanding matrix M but a dictionary (Mj : j ∈ Zs) of expanding matrices, we consider s filterbanks, one for each Mj, j ∈ Zs, defined in the same way as (12), hence
f ∗ (z) = 1, f ∗ (z) = zξk − a∗ (zMj ), j,0 j,k j,−ξk ∗ ∗ ∗ −ξk (13) gj,0(z) = aj (z), gj,k(z) = z . with ξk ∈ EMj , k 6= 0. In each of the filterbanks the number of filters is equal to | det Mj| = mj, j ∈ Zs, that is, the filterbanks are critically sampled. At each step of decompositions we choose a matrix Mj, j ∈ Zs, and we apply the corresponding analysis filters fj,k, k ∈ Zmj .
10 2.4 Shear scaling matrices In the context of shearlets, the dilatation matrices are of the form
M = DaSW , (14) where Da is a parabolic expanding anisotropic diagonal matrix 2 a Ip 0 Da = , a ∈ N, a ≥ 2, p < d, (15) 0 a Id−p and SW a shear matrix Ip W p×(d−p) SW = ,W ∈ Z . (16) 0 Id−p The shear matrix takes the role of a rotation which would be the desirable transformation which, unfortunately, neither offers a group structure in the d continuous case [?, see]]furh2015 nor keeps the grid Z invariant in the dis- crete case. Shears, on the other hand, both work in the continuous and the discrete case. The matrix Da is called a parabolic scaling matrix because its eigenvalues are squares of each other. This implies d+p det Da = a , which quickly leads to huge values when d increases. Since the determinant of Da also corresponds to the number of filters for decomposition level, a high value of this determinant is not at all desirable in applications where a reasonable depth of decomposition could only be reached starting with a very huge amount of input data. d×d Shear matrices (16) are unimodular matrices in Z , i.e. det SW = ±1. −1 Their inverses are again shear matrices SW = S−W . Moreover, they satisfy j SW = SjW and SW SW 0 = SW +W 0 . Due to these properties we have a pseudo-commuting property, namely a DaSW = SW Da which is very useful when dealing with powers of M and different scaling matrices as in the discrete shearlet transform [8]. In fact 2 2 2 a Ip 0 Ip W a Ip a W DaSW = = 0 a Id−p 0 Id−p 0 aId−p 2 Ip aW a Ip 0 a = = SaW Da = SW Da. 0 Id−p 0 aId−p
11 The minimum determinant of (14) is given by a = 2 4Ip 0 D2 = , (17) 0 2Id−p
p+d and det D2 = 2 . For the discrete shearlet transform it was proved by [8, 12, 13] that it is possible to generate a MMRA and achieve perfect reconstruction if the ac- cumulated scaling matrices
n n M = Mn ...M1 , = (1, . . . , n) ∈ {0, . . . , s − 1} =: Zs , n = ||, satisfy the following properties: all M are expanding matrices, the lattice −1 d d −1 M Z is a refinement of Z and the family (Mj : j ∈ Zs) has the slope resolution property.
Definition 7 A family (Mj : j ∈ Zs) provides the slope resolution if there exists a reference line `0 through the origin such that for any line ` there n exists a sequence ∈ Zs such that M`0 → `.
[12] has verified these three properties for the parabolic matrix (17)ina general dimension d. There, the pseudo commuting property
2 D2SW = SW D2 for parabolic matrices turned out to be useful because it allows to write explicitly the formulas for the iterated matrices
n n Y n 0 X j 0 M = D2SWj = SW D2 ,W = 2 Wj. j=1 j=1
For example, recalling that shear matrices are unimodular, the identity
−1 d −n d −n d M Z = D2 S−W 0 Z = D2 Z ,
−1 d d allows us to conclude that M Z is a refinement of Z for any n ∈ N and n ∈ Zs .
2.5 Pseudo-commuting matrices Motivated by a simple and very useful property of the scaling matrices in bivariate discrete shearlets (see [8]), the idea by [1] was to look for anisotropic
12 d×d diagonal expanding matrices D ∈ Z and unimodular matrices A, not necessarily shear matrices, such that there exist m, n ∈ N for which the commuting relationship DAm = AnD (18) is true. If two matrices A, B satisfy the requirement (18), i.e., BAm = AnB, we say that they are pseudo commuting. Of course, any pair of commuting matrices has this property. It turns out (see [1]), that this depends strongly on the dimension d. For d = 2 we have the following result.
Proposition 8 ([1]) Let D an anisotropic diagonal expanding matrix in 2×2 2×2 Z and A a non-diagonal unimodular matrix in Z . The identity
DAm = AnD is satisfied if and only if one of the following three cases holds:
q 1. A = I, and m, n such that m = `1q, n = `2q, for some `1, `2 ∈ Z; rk 0 1 w 2. D = , m, n such that r = n and A = ± ; 0 k m 0 1
rk 0 1 0T 3. D = , m, n such that r = m and A = ± . 0 k n w 1
In particular, any 2 × 2 matrix that pseudo commutes with an anisotropic scaling matrix has to be a shear matrix. For d = 3, the situation is different.
Proposition 9 ([1]) Let
krs 0 0 D = 0 kr 0 , r, s ∈ Q+, r, s 6= 1, 0 0 k and B v A = , 0 λ
2×2 2 n where B ∈ Z is unimodular and v ∈ Z . Then DA = A D holds for some n ∈ N if and only A has one of the following forms 1. v = 0, λ = ±1, B a shear matrix and s = n or λ = −1, Bn = B, s = 1 and n is odd;
13 1 w 2. v 6= 0, λ = 1 and either B = ± , s = n and r ∈ {1, n} or 0 1 1 wT B = ± , s = 1 and r ∈ {1, 1 }. 0 1 n n
1 w 3. λ = −1, n odd and either B = ± , s = n, and r ∈ {1, 1 } or 0 1 n 1 wT B = ± , s = 1 , and r ∈ {1, n}. 0 1 n
4. λ = 1, Bn = B, Bv = v, s = 1 and r = n.
A matrix B will be called n-periodic if Bn = B. Proposition 9 shows 3×3 that, in contrast to d = 2, there exists non-shear matrices in Z that pseudo–commute with anisotropic diagonal matrices. The same extends to higher dimensions.
Proposition 10 For d > 3 consider krI 0 D := p , 0 kId−p and let B W A := , 0 Id−p p×p p×d−p where B ∈ Z is a unimodular n-periodic matrix, W ∈ Z , and BW = W . Then the relation DA = AnD holds if r = n.
3 Anisotropic diagonal scaling matrices and shears
From the previous section we show that in order to generate a directional multiple multiresolution analysis we have to consider a family of jointly ex- panding matrices that satisfies the slope resolution property. Unfortunately, parabolic matrices in form (17) have a huge determinant that grows rapidly with the dimension d, so we choose here to work with an anisotropic diagonal dilatation matrix with determinant as small as possible. This is obviously the case for 3I 0 D = p . (19) 0 2Id−p One of the most important application in the shearlet context is the detec- tion of tangential hyperplanes in a point, because this permits us to catch
14 faces of a given contour. For this reason we choose p = d − 1 and we fix on a dilatation matrix of the form 3I 0 D = d−1 . (20) 0 2
As for the bivariate shearlets (see [8]), this choice is arbitrary and creates an asymmetry between the variables where the last variable is treated differ- ently from the first d−1 ones with respect to which the shearing is symmet- ric. We will also see that this prohibits the resolution of certain directions. To obtain a system that resolves all possible directions, d multiresolution systems have to be used where the special role is taken by all variables in turn. In two dimensions the difference between the parabolic matrix D2 and D from (20) is still small, det D2 = 8 and det D = 6, but when d increase the difference increases rapidly:
1d−1 det D = 22d−1, det D = 2d 1 + , 2 2 see also Figure 1.
×10 4 3.5
3
2.5
2
1.5
1
0.5
0 2 3 4 5 6 7 8
2d−1 Figure 1: Comparison between det D2 = 2 (dash line) and det D = d 1 d−1 2 1 + 2 (solid line) for d = 2,..., 8.
The special case d = 2 is discussed by [11], here we want to generalize to dimension d. We consider shear matrices of the form I −e S = d−1 j , j ∈ , (21) j 0 1 Zd
15 d−1 where ej, j ∈ Zd, are the vertices of the standard simplex in R , i.e., e0 = 0 and e1, . . . , ed−1 defined by (ej)k = δjk, k = 1, . . . , d − 1. By Proposition 10, these matrices interact with the dilation matrix D in the following way:
2 3 DSj = Sj D, j ∈ Zd. (22)
In analogy with the shearlet case in [8], we define the refinement matrices Mj, j ∈ Zd, as
3I −6e M := DS2 = d−1 j = S3D, (23) j j 0 2 j whose inverses can be easily computed as
1 −1 −2 −1 3 Id−1 ej Mj = Sj D = 1 . (24) 0 2 The multiple subdivision scheme is now governed by the matrices
n M := Mn ...M1 , = (1, . . . , n) ∈ Zd , n ∈ N, for which we can can give an explicit expression.
n Lemma 11 For n ∈ N and ∈ Zd we have
3−nI 21−np 2 M −1 = d−1 3 , (25) 0 2−n with the d − 1 multivariate polynomials
n X j−1 p (x) = x ej , x ∈ [0, 1]. j=1
Proof. Proceeding by induction on n, we first note that in the case n = 1 −1 we obtain exactly the matrices M , ∈ Zd in (24). We now suppose that the claim is true for n > 1 and verify it for n + 1 and 0 n+1 −1 = ( , n+1) ∈ Zd . The matrix M becomes
−n 1−n −1 3 I 2 p 0 (2/3) 3 e M −1 = M −1M −1 = d−1 n+1 0 n+1 0 2−n 0 2−1 −(n+1) −n −n 3 I 2 p 0 (2/3) + 3 e = d−1 n+1 , 0 2−(n+1)
16 and taking into account that n −n 2 −n −n 2 2 −n 2 2 p 0 + 3 e = 2 p 0 + e = 2 p 3 n+1 3 3 n+1 3 we get (25). Inverting (25), we also obtain the general expression for M, namely n 3nI 3n+1 q (2/3) X M = d−1 , q (x) := − xje , ∈ n, n ∈ . 0 2n j Zd N j=1 (26) Now we verify some properties of M that imply that these family of scal- ing matrices is useful for a directional multiresolution analysis based on −1 d d an MMRA: they are jointly expanding, i.e., the grid M Z tends to R uniformly and they satisfy a slope resolution property.
Proposition 12 (Mj : j ∈ Zd) are jointly expanding matrices.
Proof. We have to show that the joint spectral radius of the scaling matrices
−1 1/n ρ (Mj : j ∈ d) = lim max M , Z n→∞ n ∈Zd defined by [10], is strictly less than one. By the equivalence of norms onfinite dimensional spaces, we can choose the underlying matrix norm arbitrarily. Using k·k1, we obtain for n ∈ N that
−n 1−n 1/n −1 1/n 3 2 p (2/3) max M = max −n ∈ n 1 ∈ n 0 2 Zd Zd 1 1/n 1−n 2 −n = max 2 p + 2 ∈ n Zd 3 1 1/n 1/n n−1 j ∞ j 1 X 2 1 X 2 ≤ 2 + 1 ≤ 2 + 1 2 3 2 3 j=0 j=0 71/n = 2
−1 1/n max n M < 1 n ≥ 3 and therefore ∈Zd 1 for independently of . n For each ∈ Zd , we can rewrite (26) as 3nI 3n+1q (2/3) I 3q (2/3) M = d−1 = Dn d−1 =: DnS (27) 0 2n 0 1
17 and (25) as
n−j ! 3−nI 21−np (2/3) 3 Pn 3 e M −1 = d−1 = D−n Id−1 j=1 2 j 0 2−n 0 1 −n =: D Sb. (28) Taking inverses of (27) and (28) also yields
−1 n −1 −1 −n M = Sb D ,M = S D . (29) −1 These formulations of M and M allow us to compute efficiently the grids d −1 d n −n MZ and M Z as transformations of the D or D refined grid. These transformations are still shears, however, in contrast to the shearlet case with the parabolic matrix D2, they have no integer translations any more. d T d Given a hyperplane H = {x ∈ R : v x = 0} ⊂ R whose normal has the T property vd 6= 0, we can renormalize v such that vd = 1. Writing v = (t, 1) , d−1 we call t ∈ R the slope of the hyperplane H. The next result shows that any such hyperplane can be obtained, up to arbitrary precision, from a single reference hyperplane by applying a proper matrix M. This property is called slope resolution property and ensures that an anisotropic transform which captures singularities across the reference hyperplane can capture directional singularities across arbitrary hyperplanes in a MMRA, where the modified direction can be directly read off fromthe index . The slope of the reference hyperplane can even be chosen arbitrarily in a scaled version of the standard simplex
( d−1 ) d−1 X ∆d−1 := t ∈ R : ti ≥ 0, ti ≤ 1 . i=1
Theorem 13 For any reference hyperplane H with slope t ∈ 6∆d−1, the matrix family (Mj : j ∈ Zd) has the slope resolution property: for any 0 0 d−1 hyperplane H with slope t ∈ R and any δ > 0 there exists : N → Zd such that 0 t −n t − 2 M < δ. 1 1
0 0 d−1 Proof. Let H be any hyperplane with slope t ∈ R . A multiplication −1 with Mj , j ∈ Zd, changes the slope in the following way: 1 2 −1 t 3 Id−1 ej t 1 3 t + 2ej d−1 Mj = 1 = , t ∈ R , 1 0 2 1 2 1
18 (0,6)
h (0,4) 2
(0,2) h h 0 1
(0,0)(2,0) (4,0) (6,0)
Figure 2: In two dimension (d − 1 = 2), we have three different functions hj, j ∈ {0, 1, 2} and the union of h0(6∆2) (red), h1(6∆2) (green), h2(6∆2) (orange) gives 6∆2.
d−1 d−1 due to which we define the affine contractions hj : R → R by 2 h (t) := t + 2e , j ∈ , (30) j 3 j Zd which satisfy 2 h (6∆ ) = 6∆ + 2e = 4∆ + 2e ⊂ 6∆ (31) j d−1 3 d−1 j d−1 j d−1 and [ hj(6∆d−1) = 6∆d−1, (32) j∈Zd as visualized in Figure 2 for d = 3. As pointed out by [5], the compact set 6∆d−1 is an invariant space with respect to the contractions hj, j ∈ Zd and d−1 for any compact subset X ⊂ R the generating property [ 6∆d−1 = lim h(X) (33) n→∞ n ∈Zd
n holds true in the Hausdorff metric. For general ∈ Zd we have
−n 1−n 2 2 n 2 t 3 I 2 p t t + 2p M −1 = d−1 3 = 2−n 3 3 , 1 0 2−n 1 1 which involves the affine contractions 2n 2 h (t) = h h (t) = t + 2p , ∈ n, n ∈ . n (1,...,n−1) 3 3 Zd N
19 This iterative definition and (31) yield that n h(6∆d−1) ⊂ 6∆d−1, ∈ Zd , n ∈ N. (34) 0 d−1 0 For given t ∈ R , and 0 < δ < δ, we first consider the compact set 0 d−1 0 0 Rδ0 (t ) = {u ∈ R : kt − uk1 ≤ δ }, for which the generating property 1 n1 (33) implies that there exist n1 ∈ N and ∈ Zd such that 0 I := h1 Rδ0 (t ) ∩ 6∆ 6= ∅,
Again by (33), this time applied to the closure of I, there exist n2 ∈ N and 2 n2 ∈ Zd such that 0 0 2 1 n t ∈ h2 (I) ⊂ h2 h1 Rδ0 (t ) =: h Rδ0 (t ) , = ( , ) ∈ Zd , (35) −1 0 hence h (t) ∈ Rδ0 (t ) or, equivalently, 0 t −n t 0 − 2 M ≤ δ < δ 1 1 as claimed.
4 Some numerical examples
Finally, we illustrate how these matrices work as a MMRA, see Section 2.2, by providing some explicit examples and applying them to images. To that end, we focus on the two dimensional case, where we consider the family of matrices 3 0 3 −6 M = ,M = , (36) 0 0 2 1 0 2 with det(Mj) = 6, for j ∈ Z2. This means that we have to provide 6 analysis filters and 6 synthesis filters for each Mj, j ∈ Z2. The Smith factorization form of the matrices is 3 −6 1 −3 3 0 1 0 M = IDI,M = = = S3DI. 0 1 0 2 0 1 0 2 0 1 (37) Following the construction presented in Section 2.2, we start with two uni- variate interpolatory subdivision schemes with scaling factors σ1 = 3 and σ2 = 2, respectively, like the ternary and dyadic piecewise linear interpola- tory schemes with masks 1 2 2 1 1 1 b = ..., 0, , , 1, , , 0,... , b = ..., 0, , 1, , 0,... . 1 3 3 3 3 2 2 2
20 Using (11), the symbols of the subdivision schemes related to the matrices M0 and M1 are
z−2z−1 a∗(z) = b∗ (z) = 1 2 (1 + z + z2)2(1 + z )2, (38) 0 D 6 1 1 2 −1 3 z1z a∗(z) = b∗ (zS ) = 2 (1 + z + z2)2(1 + z−3z )2. (39) 1 D 6 1 1 1 2 Once we have the subdivision schemes for both the matrices we can com- pute the filters using the prediction-correction method and (13). Then, the analysis and synthesis filters for the diagonal matrix M0 are
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 2 f0,0 = 0 0 1 0 0 , f0,1 = 0 − 1 0 − , f0,2 = 0 − 0 1 − , 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 0 0 − 2 0 0 0 − 3 0 0 − 6 0 − 6 0 0 − 3 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 1 1 f0,3 = 0 0 − 0 0 , f0,4 = 0 − 0 0 − , f0,5 = 0 − 0 0 − 2 3 6 6 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 6 3 2 3 6 0 0 0 0 0 0 0 0 0 0 1 2 2 1 g0,0 = 3 3 1 3 3 , g0,1 = 0 1 0 0 0 , g0,2 = 1 0 0 0 0 , 1 1 1 1 1 6 3 2 3 6 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g0,3 = 0 0 0 0 0 , g0,4 = 0 0 0 0 0 , g0,5 = 0 0 0 0 0 , 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 where the boldface entry highlights the (0, 0) element. In the same way we
21 c
M0 M1
1 1 1 1 c0 d0 c1 d1
M0 M M 1 M0 1
2 2 c2 d2 c2 d2 c2 d2 c00 d00 01 01 10 10 11 11
Figure 3: Multiple decomposition tree up to level 2.
obtain the analysis and synthesis filters for the matrix M1,
0 0 0 0 0 0 0 0 0 0 0 2 1 f1,0 = 0 0 1 0 0 , f1,1 = 0 − 3 1 0 − 3 0 , 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 − 2 0 0 0 0 0 0 1 2 f1,2 = 0 − 3 0 1 − 3 , f1,3 = 0 0 0 1 0 0 0 , 1 0 0 0 0 0 0 0 0 0 0 0 − 2
1 1 − 3 0 0 − 6 0 0 0 0 0 0 f1,4 = 0 0 0 0 1 0 0 0 0 0 , 1 1 0 0 0 0 0 0 − 3 0 0 − 6 1 1 − 6 0 0 − 3 0 0 0 0 0 0 f1,5 = 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 − 6 0 0 − 3
1 1 1 1 1 6 3 2 3 6 0 0 0 0 0 0 0 0 0 0 0 1 2 2 1 g1,0 = 0 0 0 3 3 1 3 3 0 0 0 , g1,1 = 0 1 0 0 0 , 1 1 1 1 1 0 0 0 0 0 0 6 3 2 3 6 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 g1,2 = 1 0 0 0 0 , g1,3 = 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 g1,4 = 0 0 0 0 0 , g1,5 = 0 0 0 0 0 . 0 0 0 0 1 0 0 0 1 0
In each step of the analysis process we can now choose either M0 or M1. After two decomposition levels we thus have 4 different decompositions, as depicted by the tree in Figure 3. If we apply two levels of decompositions on
22 Figure 4: Two level decomposition of the image in Figure 6a. The six images 2 2 in each row are the scaling coefficients c and details coefficients d,k with ∈ Z2 and k = 1,..., 5. Each row represents a branch of the tree in Figure 3 from = (0, 0) to (1, 1).
Figure 5: Two level decomposition of the image in Figure 6a using Daubechies wavelet of order 2. The 4 images are the scaling coefficients c2 and details coefficients along the horizontal, vertical, diagonal direction, from left to right. the test image in Figure 6a, then the resulting scaling and details coefficients are displayed in Figure 4. The effect of the matrix M1, or better of the shear, on the decomposition coefficients, is to shift the dominant directions for edges in the image. The more applications of M1 we do, the more the resulting image is “rotated” and stretched, as shown in the last row of Figure 4. As for wavelets, each detail coefficient detects features along a certain direc- tion. In the tensor product wavelet case these relevant directions are only the horizontal, the vertical and the diagonal one, see Figure 5. In our case the combination of the details directions ξk with the shearing of the images allows us to detect different directional features in different branches ofthe tree. One can analyze this behavior looking at Figures 6 and 7 where, after
23 two levels and following all the different branches separately, we reconstruct only the details while removing the scaling coefficients. We recall that the detail coefficients are higher along the directions considered, this clearly shows which directions are considered for each different branch. Due to the tree structure, the analysis with shearlets is highly redundant. This could be a disadvantage from the point of view of memory consump- tion but keeping only the well approximated parts from each branch will eventually have to potential for very efficient compression. In fact once we have computed the full decomposition tree, we can reconstruct the original image either following a single branch or combining together the branches that emerge from the same knot. In the second case, every time we re- construct one level we take some average between the reconstructed images that refer to the same knot in the tree. In Figures 6 and 7, we compare the reconstructed details following a specific branch and combining all the branches, see Figures 6b and 7b. The reconstructed details as mean of all the branches detects more features than the reconstruction along a single branch. Moreover, taking the mean of all the branches mitigates the arti- facts that occur in the single branch. For example, the double application of M1 shows some blur on the edges of the image, see the bottom right images in Figures 6 and 7. This effect is less visible if we reconstruct averaging all the branches of the tree, Figures 6b and 7b.
5 Conclusion
We proposed a shearlet-like decomposition family of matrices that allow to define a directional transform, based on shears and a nonparabolic scaling matrix. These matrices are jointly expanding, thus suitable for a multiple multiresolution process. In particular, we show how to define a convergent multiple subdivision scheme and the respective filterbanks using this family of matrices. In any dimension d, we have also proved that they satisfy the slope resolution property, crucial for having a tool capable to capture the directional features of a signal in arbitrary directions. Moreover, the proposed matrices satisfy the pseudo commuting property that allows a faster computation of the refined (coarse) lattice. In contrast to classical shearlets, the matrices considered here have smaller determinant. This is a useful feature because the determinant gives the number of filters to be considered and so it is strictly related to the compu- tational complexity of the implementation. We have also illustrated by means of a few examples how this family of ma-
24 (a) Original image (b) All the tree
(c) = (0, 0) (d) = (0, 1)
(e) = (1, 0) (f) = (1, 1)
Figure 6: Reconstructed details following different branches of the tree (Fig. 3) after two levels of decomposition.
25 (a) Original image (b) All the tree
(c) = (0, 0) (d) = (0, 1)
(e) = (1, 0) (f) = (1, 1)
Figure 7: Reconstructed details, after two levels of decompositions, following different branches of the tree or considering all the tree together.
26 trices acts on images and the potential of using it. In fact, as expected, the details of the images capture the discontinuities along different directions.
Acknowledgements
This research has been accomplished within Rete ITaliana di Approssi- mazione (RITA).
References
[1] Bozzini, M., Ghisi, D., Rossini, M. & Sauer, T. (2015) Direc- tional transforms and pseudo-commutating properties. Monografías Matemáticas, 40, 29–41.
[2] Cotronei, M., Ghisi, D., Rossini, M. & Sauer, T.(2015) An anisotropic directional subdivision and multiresolution scheme. Ad- vances in Computational Mathematics, 41, 709–726.
[3] Fell, J., Führ, H. & Voigtlaender, F.(2015) Resolution of the wave front set using general wavelet transforms. Proceeding of Sam- pling Theory and Applications (SampTA) 2015. Washington: IEEE, pp. 332–336.
[4] Hamming, R. W. (1998) Digital filters. New York: Dover Publications.
[5] Hutchinson, J. E. Fractals and self-similarity. Indiana University Journal of Math, 30, 713–747.
[6] Labate, D., Lim, W-Q., Kutyniok, G. & Weiss, G. Sparse mul- tidimensional representation using shearlets. Optics & Photonics 2005, 5914.
[7] Kutyniok, G., Labate, D.(2011) Shearlets: Multiscale Analysis for Multivariate Data. New York: Springer Science & Business Media.
[8] Kutyniok, G. & Sauer, T.(2009) Adaptive directional subdivision schemes and shearlet multiresolution analysis. SIAM Journal on Math- ematical Analysis, 41, 1436–1471.
[9] Marcus, M. & Minc, H.(1969) A survey of matrix theory and matrix inequalities. Boston: Prindle, Weber & Schmidt.
27 [10] Rota, G. C. & Strang, G.(1960) A note on the joint spectral radius. Proceedings of the Netherlands Academy, 22, pp. 379–381,
[11] Rossini, M. & Volontè, E.(2017) On directional scaling matrices in dimension d = 2. Mathematics and Computers in Simulation. (in press).
[12] Sauer, T. Shearlet multiresolution and multiple refinement. Shearlets (G. Kutyniok & D. Labate eds.) New York: Springer Science & Business Media, pp. 199–237.
[13] Sauer, T.(2012) Multiple subdivision. Proceeding of Curves and Sur- faces 2011 (J.-D. Boissonnat et al. eds.). Berlin: Springer, pp. 612–628.
[14] Vetterli, M. & Kovačević, J. (1995) Wavelets and subband coding. New Jersey: Prentice Hall.
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